Simple traversable wormholes violating energy conditions only near the Planck scale

Abstract

We present a static and axisymmetric traversable wormhole spacetime with vanishing Arnowitt-Deser-Misner (ADM) mass which is characterized by a length parameter l and a deformation parameter a and reduces to the massless Kerr vacuum wormhole as l → 0. The spacetime is analytic everywhere and regularizes a ring-like conical singularity of the massless Kerr wormhole by virtue of a localized exotic matter which violates the standard energy conditions only near the wormhole throat. In the spherically symmetric case (a = 0), the areal radius of the wormhole throat is exactly l and all the standard energy conditions are respected outside the proper radial distance approximately 1.60l from the throat. While the curvature at the throat is beyond the Planck scale if l is identical to the Planck length l p, our wormhole may be a semi-classical model for l ≃ 10l p. With l = 10l p, the total amount of the negative energy supporting this wormhole is only E ≃ −26.5m p c 2, which is the rest mass energy of about −5.77 × 10−4 g. It is shown that the geodesic behavior on the equatorial plane does not qualitatively change by the localization of an exotic matter.